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Theorem enrefg 6758
Description: Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
enrefg (𝐴𝑉𝐴𝐴)

Proof of Theorem enrefg
StepHypRef Expression
1 f1oi 5495 . . 3 ( I ↾ 𝐴):𝐴1-1-onto𝐴
2 f1oen2g 6749 . . 3 ((𝐴𝑉𝐴𝑉 ∧ ( I ↾ 𝐴):𝐴1-1-onto𝐴) → 𝐴𝐴)
31, 2mp3an3 1326 . 2 ((𝐴𝑉𝐴𝑉) → 𝐴𝐴)
43anidms 397 1 (𝐴𝑉𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148   class class class wbr 4000   I cid 4285  cres 4625  1-1-ontowf1o 5211  cen 6732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-en 6735
This theorem is referenced by:  enref  6759  eqeng  6760  domrefg  6761  mapdom1g  6841  fidifsnen  6864  nnfi  6866  onenon  7177  oncardval  7179  cardonle  7180  dju1en  7206  xpdjuen  7211  iseqf1olemqf1o  10476  hashun  10766
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